Find the value of $ \lim_{n\rightarrow\infty}\sqrt[n]{\int_{1}^{3}x^{n/x} dx}$

Asked Jul 19 '20 at 07:21

Active Jul 19 '20 at 16:10

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Find the value of $ \lim_{n\rightarrow\infty}\sqrt[n]{\int_{1}^{3}x^{n/x} dx}$.

My attempt:

$$ L = (\lim_{x \to \infty}\int_{1}^{3} x^{\frac {n}{x}}\mathrm{d}x)^\frac {1} {n} \Longrightarrow \log L = \frac {\lim_{x \to \infty}\int_{1}^{3} x^{\frac{n}{x}}\mathrm{d}x} {n}$$

Using L'H$\hat{\mathrm{o}}$pital's rule,

$ \Longrightarrow \log L = \frac {\lim_{x \to \infty}\frac {\int_{1}^{3} \log x x^{\frac{n}{x}} \mathrm{d}x} {x}} {\lim_{x \to \infty}\int_{1}^{3} x^{\frac{n}{x}}\mathrm{d}x} = \frac {L \log 3} {3} $

Comparing $\frac {\log 3}{3} = \frac {\log L}{L}$,

$ L = 3$.

But the given answer is $ e^{1/e}$.

edited Jul 19 '20 at 16:10

Eric Monlye

asked Jul 19 '20 at 07:21

Anonymous

You are mixing up variables. $x$ is a variable of integration and the limit is not w.r.t. $x$. – Kavi Rama Murthy Jul 19 '20 at 07:32
3

Does this answer your question? If $f(x)$ is continuous on $[a,b]$ and $M=\max ; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n,\mathrm dx\right)^{1/n}$? – metamorphy Jul 19 '20 at 07:36

Find the value of $ \lim_{n\rightarrow\infty}\sqrt[n]{\int_{1}^{3}x^{n/x} dx}$

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